= Solution
Suppose ZFC proved that every $\aleph_1$-Suslin set is determined. The assumed consistency of ZFC and the relative consistency of the <Continuum hypothesis> would then give a model of
$$
\mathrm{ZFC}+\mathsf{CH}+\text{“every $\aleph_1$-Suslin set is determined”.}
$$
In that model $2^{\aleph_0}=\aleph_1$. By <Every set of reals is continuum-Suslin>, every subset of $\omega^\omega$ is therefore $\aleph_1$-Suslin and hence determined. This is the <axiom of determinacy>.
But the <axiom of choice> produces an undetermined set of reals, so ZFC and the axiom of determinacy are incompatible. The displayed theory cannot have a model, contradicting the relative consistency of ZFC plus the continuum hypothesis. Hence ZFC cannot prove that all $\aleph_1$-Suslin sets are determined.
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